N-dimensional discrete Fourier transform via bosonic Hamiltonian
Abstract
The discrete Fourier transform (DFT) underpins many classical algorithms and is a fundamental unitary operator for quantum information processing. Implementing the N-dimensional DFT in photonic integrated circuits (PICs) is limited by the cascades of Mach-Zehnder interferometers that current architectures require. Here we propose bosonic Hamiltonians that realize the N-dimensional DFT through a single stage of multimode evolution, complemented only by phase shifters before and after the interaction region, in a geometry suited to 3D waveguides. Modeling the system as a graph, where edges correspond to couplings and the vertices are the waveguides, we obtain analytical solutions for complete graphs up to N=6 and numerical solutions up to N=31. For non-complete graphs, different propagation constants are required in the Hamiltonian. We report all solutions for N≤ 8, partial exploration for N=9, and selected cases for N=10, together with three conjectures that guide the numerical search for N ≥ 11. These configurations circumvent the vanishing evanescent coupling strength imposed by the waveguide separation, and a closed-form sensitivity criterion selects those that are admissible as a waveguide layout and least sensitive to fabrication error. We also uncover the missing non-affine parameters of the $6$-dimensional DFT, and show that the scaling law for implementing the N-dimensional DFT with our building blocks is 𝒪(NloglogN). This allows assembling the $2520$-dimensional DFT with only $2625$ interferometers, in contrast to the ≈ 3× 106 of Reck and Clements architectures.
Related
- Design and characteristics of a cavity-enhanced Fourier-transform spectrometer based on a supercontinuum source
- High-speed optical comb-based 3D tomographic profiling using Fourier-transform method with reduced-frame acquisition
- When is a bosonic operator a Hamiltonian?
- Fourier Transform-Based Computation of Two-Electron Four-Center Coulomb Repulsion Integrals
- Quantum Fourier transform for the symmetric group
BibTeX
@misc{barriga2026ndimensional,
title = {{$N$-dimensional discrete Fourier transform via bosonic Hamiltonian}},
author = {Edgar Barriga and Camila Muñoz and Alejandro Muñoz and Santiago Rojas-Rojas and Pablo Solano and Carla Hermann-Avigliano and Aldo Delgado},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05644},
eprint = {2609.05644},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05644},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - N-dimensional discrete Fourier transform via bosonic Hamiltonian
AU - Edgar Barriga
AU - Camila Muñoz
AU - Alejandro Muñoz
AU - Santiago Rojas-Rojas
AU - Pablo Solano
AU - Carla Hermann-Avigliano
AU - Aldo Delgado
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.05644
UR - https://arxiv.org/abs/2609.05644
AB - The discrete Fourier transform (DFT) underpins many classical algorithms and is a fundamental unitary operator for quantum information processing. Implementing the N-dimensional DFT in photonic integrated circuits (PICs) is limited by the cascades of Mach-Zehnder interferometers that current architectures require. Here we propose bosonic Hamiltonians that realize the N-dimensional DFT through a single stage of multimode evolution, complemented only by phase shifters before and after the interaction region, in a geometry suited to 3D waveguides. Modeling the system as a graph, where edges correspond to couplings and the vertices are the waveguides, we obtain analytical solutions for complete graphs up to N=6 and numerical solutions up to N=31. For non-complete graphs, different propagation constants are required in the Hamiltonian. We report all solutions for N≤ 8, partial exploration for N=9, and selected cases for N=10, together with three conjectures that guide the numerical search for N ≥ 11. These configurations circumvent the vanishing evanescent coupling strength imposed by the waveguide separation, and a closed-form sensitivity criterion selects those that are admissible as a waveguide layout and least sensitive to fabrication error. We also uncover the missing non-affine parameters of the $6$-dimensional DFT, and show that the scaling law for implementing the N-dimensional DFT with our building blocks is 𝒪(NloglogN). This allows assembling the $2520$-dimensional DFT with only $2625$ interferometers, in contrast to the ≈ 3× 10^6 of Reck and Clements architectures.
ER -